Extruding inwards leaves vertices alone
Yes, it does.

Extruding in 3D modeling means something like:


  to shift a selection in space while creating all the geometry
  that is needed to keep all the non-selected parts in place


You can have more detailed information about how tha algorithm works in the official manual page.



Let's consider a UV sphere:

 

Notice how in the example above the non selected faces keep their positions (and so their edges and vertices). Is something different from grabbing, as involves the creation of new geometry.

 

By extruding, for example, along the normal axis, you may see from the pictures above how the operation does the same thing. In both cases (positive and negative normal axis extrusion) the surrounding vertices of the initial selection geometry keep their position.



One way to achieve the result you want is to extrude inward and than dissolve the starting geometry.




After inward extrusion, assign the selected faces to a vertex group, than extend the selection with Ctrl++. Than that go to Vertex groups and remove previuosly assinged vertex group from the selection.



Press X to open the "Delete" menue and choose "Dissolve edges".